# Lehmann–Scheffé theorem

Lehmann–Scheffé theorem This article needs additional citations for verification. Ajude a melhorar este artigo adicionando citações a fontes confiáveis. O material sem fonte pode ser contestado e removido. Encontrar fontes: "Lehmann–Scheffé theorem" – notícias · jornais · livros · acadêmico · JSTOR (abril 2011) (Saiba como e quando remover esta mensagem de modelo) In statistics, the Lehmann–Scheffé theorem is a prominent statement, tying together the ideas of completeness, suficiência, uniqueness, and best unbiased estimation.[1] The theorem states that any estimator which is unbiased for a given unknown quantity and that depends on the data only through a complete, sufficient statistic is the unique best unbiased estimator of that quantity. The Lehmann–Scheffé theorem is named after Erich Leo Lehmann and Henry Scheffé, given their two early papers.[2][3] If T is a complete sufficient statistic for θ and E(g(T)) = τ(eu) then g(T) is the uniformly minimum-variance unbiased estimator (UMVUE) of τ(eu).

Conteúdo 1 Declaração 1.1 Prova 2 Example for when using a non-complete minimal sufficient statistic 3 Veja também 4 References Statement Let {estilo de exibição {vec {X}}=X_{1},X_{2},pontos ,X_{n}} be a random sample from a distribution that has p.d.f (or p.m.f in the discrete case) {estilo de exibição f(x:teta )} Onde {displaystyle theta in Omega } is a parameter in the parameter space. Suponha {displaystyle Y=u({vec {X}})} is a sufficient statistic for θ, e deixar {estilo de exibição {f_{S}(y:teta ):theta in Omega }} be a complete family. Se {estilo de exibição varphi :nome do operador {E} [varphi (S)]=theta } então {estilo de exibição varphi (S)} is the unique MVUE of θ.

Proof By the Rao–Blackwell theorem, E se {estilo de exibição Z} is an unbiased estimator of θ then {estilo de exibição varphi (S):=nome do operador {E} [Zmid Y]} defines an unbiased estimator of θ with the property that its variance is not greater than that of {estilo de exibição Z} .

Now we show that this function is unique. Suponha {estilo de exibição W.} is another candidate MVUE estimator of θ. Then again {estilo de exibição psi (S):=nome do operador {E} [Wmid Y]} defines an unbiased estimator of θ with the property that its variance is not greater than that of {estilo de exibição W.} . Então {nome do operador de estilo de exibição {E} [varphi (S)-psi (S)]=0,theta in Omega .} Desde {estilo de exibição {f_{S}(y:teta ):theta in Omega }} is a complete family {nome do operador de estilo de exibição {E} [varphi (S)-psi (S)]=0implies varphi (y)-psi (y)=0,theta in Omega } and therefore the function {estilo de exibição varphi } is the unique function of Y with variance not greater than that of any other unbiased estimator. We conclude that {estilo de exibição varphi (S)} is the MVUE.

Example for when using a non-complete minimal sufficient statistic An example of an improvable Rao–Blackwell improvement, when using a minimal sufficient statistic that is not complete, was provided by Galili and Meilijson in 2016.[4] Deixar {estilo de exibição X_{1},ldots ,X_{n}} be a random sample from a scale-uniform distribution {displaystyle Xsim U((1-k)teta ,(1+k)teta ),} with unknown mean {nome do operador de estilo de exibição {E} [X]=theta } and known design parameter {parentes de estilo de exibição (0,1)} . In the search for "best" possible unbiased estimators for {estilo de exibição teta } , it is natural to consider {estilo de exibição X_{1}} as an initial (crude) unbiased estimator for {estilo de exibição teta } and then try to improve it. Desde {estilo de exibição X_{1}} is not a function of {displaystyle T=left(X_{(1)},X_{(n)}certo)} , the minimal sufficient statistic for {estilo de exibição teta } (Onde {estilo de exibição X_{(1)}=min _{eu}X_{eu}} e {estilo de exibição X_{(n)}=máximo _{eu}X_{eu}} ), it may be improved using the Rao–Blackwell theorem as follows: {estilo de exibição {chapéu {teta }}_{RB}=nome do operador {E} _{teta }[X_{1}mid X_{(1)},X_{(n)}]={fratura {X_{(1)}+X_{(n)}}{2}}.} No entanto, the following unbiased estimator can be shown to have lower variance: {estilo de exibição {chapéu {teta }}_{LV}={fratura {1}{k^{2}{fratura {n-1}{n+1}}+1}}cdot {fratura {(1-k)X_{(1)}+(1+k)X_{(n)}}{2}}.} And in fact, it could be even further improved when using the following estimator: {estilo de exibição {chapéu {teta }}_{texto{BAYES}}={fratura {n+1}{n}}deixei[1-{fratura {{fratura {X_{(1)}(1+k)}{X_{(n)}(1-k)}}-1}{deixei({fratura {X_{(1)}(1+k)}{X_{(n)}(1-k)}}certo)^{n+1}-1}}certo]{fratura {X_{(n)}}{1+k}}} The model is a scale model. Optimal equivariant estimators can then be derived for loss functions that are invariant.[5] See also Basu's theorem Complete class theorem Rao–Blackwell theorem References ^ Casella, Jorge (2001). Statistical Inference. Duxbury Press. p. 369. ISBN 978-0-534-24312-8. ^ Lehmann, E. EU.; Scheffé, H. (1950). "Completeness, similar regions, and unbiased estimation. EU." Sankhyā. 10 (4): 305-340. doi:10.1007/978-1-4614-1412-4_23. JSTOR 25048038. MR 0039201. ^ Lehmann, E.L.; Scheffé, H. (1955). "Completeness, similar regions, and unbiased estimation. II". Sankhyā. 15 (3): 219-236. doi:10.1007/978-1-4614-1412-4_24. JSTOR 25048243. MR 0072410. ^ Tal Galili & Isaac Meilijson (31 Mar 2016). "An Example of an Improvable Rao–Blackwell Improvement, Inefficient Maximum Likelihood Estimator, and Unbiased Generalized Bayes Estimator". The American Statistician. 70 (1): 108-113. doi:10.1080/00031305.2015.1100683. PMC 4960505. PMID 27499547. ^ Taraldsen, Gunnar (2020). "Micha Mandel (2020), "The Scaled Uniform Model Revisited," The American Statistician, 74:1, 98–100: Comente". The American Statistician. 74 (3): 315–315. doi:10.1080/00031305.2020.1769727. hide vte Statistics OutlineIndex show Descriptive statistics show Data collection hide Statistical inference Statistical theory PopulationStatisticProbability distributionSampling distribution Order statisticEmpirical distribution Density estimationStatistical model Model specificationLp spaceParameter locationscaleshapeParametric family Likelihood (monotone)Location–scale familyExponential familyCompletenessSufficiencyStatistical functional BootstrapUVOptimal decision loss functionEfficiencyStatistical distance divergenceAsymptoticsRobustness Frequentist inference Point estimation Estimating equations Maximum likelihoodMethod of momentsM-estimatorMinimum distanceUnbiased estimators Mean-unbiased minimum-variance Rao–BlackwellizationLehmann–Scheffé theoremMedian unbiasedPlug-in Interval estimation Confidence intervalPivotLikelihood intervalPrediction intervalTolerance intervalResampling BootstrapJackknife Testing hypotheses 1- & 2-tailsPower Uniformly most powerful testPermutation test Randomization testMultiple comparisons Parametric tests Likelihood-ratioScore/Lagrange multiplierWald Specific tests Z-test (normal)Student's t-testF-test Goodness of fit Chi-squaredG-testKolmogorov–SmirnovAnderson–DarlingLillieforsJarque–BeraNormality (Shapiro–Wilk)Likelihood-ratio testModel selection Cross validationAICBIC Rank statistics Sign Sample medianSigned rank (Wilcoxon) Hodges–Lehmann estimatorRank sum (Mann–Whitney)Nonparametric anova 1-way (Kruskal–Wallis)2-caminho (Friedman)Ordered alternative (Jonckheere–Terpstra)Van der Waerden test Bayesian inference Bayesian probability priorposteriorCredible intervalBayes factorBayesian estimator Maximum posterior estimator show CorrelationRegression analysis show Categorical / Multivariate / Time-series / Survival analysis show Applications Category Mathematics portalCommons WikiProject Categories: Theorems in statisticsEstimation theory

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